Proof

Proof. Induct on unnormalized branch states. Complete instruments include all outcomes and preserve total probability when summed. Input-independent random setting selection is included in the history or aligned by the same seed.

For the test,

The maximum of $E_Q\phi -E_P\phi$ is total variation, attained by the indicator of outcomes where $Q>P$. Identical independent repeated-run laws have identical product laws. More generally, the same conclusion holds whenever the complete joint repeated-run laws are equal.